Question
Rewrite the parametric equations
y=ln(x2−1+∣x∣)y=ln(−x2−1+∣x∣)
Evaluate
{x=t2+1y=ln(t+t2+1)
Choose the parametric equation
x=t2+1
Solve the equation
t=x2−1t=−x2−1
Substitute the given value of t=x2−1 into the equation y=ln(t+t2+1)
y=ln(x2−1+∣x∣)t=−x2−1
Solution
y=ln(x2−1+∣x∣)y=ln(−x2−1+∣x∣)
Show Solution
Find the first derivative
dxdy=t1
Evaluate
{x=t2+1y=ln(t+t2+1)
To find the derivative dxdy,first find dtdx and dtdy
dtd(x)=dtd(t2+1)dtd(y)=dtd(ln(t+t2+1))
Find the derivative
More Steps
Evaluate
dtd(x)=dtd(t2+1)
Calculate the derivative
More Steps
Evaluate
dtd(x)
Use differentiation rules
dxd(x)×dtdx
Use dxdxn=nxn−1 to find derivative
dtdx
dtdx=dtd(t2+1)
Calculate the derivative
More Steps
Evaluate
dtd(t2+1)
Use the chain rule dxd(f(g))=dgd(f(g))×dxd(g) where the g=t2+1, to find the derivative
dgd(g)×dtd(t2+1)
Find the derivative
2g1×dtd(t2+1)
Find the derivative
2g1×2t
Substitute back g=t2+1
2t2+11×2t
Cancel out the common factor 2
t2+11×t
Multiply the terms
t2+1t
dtdx=t2+1t
dtdx=t2+1tdtd(y)=dtd(ln(t+t2+1))
Find the derivative
More Steps
Evaluate
dtd(y)=dtd(ln(t+t2+1))
Calculate the derivative
More Steps
Evaluate
dtd(y)
Use differentiation rules
dyd(y)×dtdy
Use dxdxn=nxn−1 to find derivative
dtdy
dtdy=dtd(ln(t+t2+1))
Calculate the derivative
More Steps
Evaluate
dtd(ln(t+t2+1))
Use the chain rule dxd(f(g))=dgd(f(g))×dxd(g) where the g=t+t2+1, to find the derivative
dgd(ln(g))×dtd(t+t2+1)
Use dxdlnx=x1 to find derivative
g1×dtd(t+t2+1)
Calculate
g1×t2+1t2+1+t
Substitute back
t+t2+11×t2+1t2+1+t
Cancel out the common factor t2+1+t
t2+11
dtdy=t2+11
dtdx=t2+1tdtdy=t2+11
Find the required derivative by Substituting dtdx=t2+1t and dtdy=t2+11 into dxdy=dtdxdtdy
dxdy=t2+1tt2+11
Solution
More Steps
Evaluate
t2+1tt2+11
Multiply by the reciprocal
t2+11×tt2+1
Cancel out the common factor t2+1
t1
dxdy=t1
Show Solution